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REMARKS CONCERNING FOURIER'S THEOREM AS APPLIED TO PHYSICAL PROBLEMS.
[Philosophical Magazine, Vol. xxiv. pp. 864 — 869, 1912.]
FOURIER'S theorem is of great importance in mathematical physics, but difficulties sometimes arise in practical applications which seem to have their origin in the aim at too great a precision. For example, in a series of observations extending over time we maybe interested in what occurs during seconds or years, but we are not concerned with and have no materials for a remote antiquity or a distant future ; and yet these remote times determine whether or not a period precisely defined shall be present. On the other hand, there may be no clearly marked limits of time indicated by the circumstances of the case, such as would suggest the other form of Fourier's theorem where everything is ultimately periodic. Neither of the usual forms of the theorem is exactly suitable. Some method of taking off the edge, as it were, appears to be called for.
The considerations which follow, arising out of a physical problem, have cleared up my own ideas, and they may perhaps be useful to other physicists.
A train of waves of length X, represented by
advances with velocity c in the negative direction. If the medium is absolutely uniform, it is propagated without disturbance ; but if the medium is subject to small variations, a reflexion in general ensues as the waves pass any' place x. Such reflexion reacts upon the original waves; but if we suppose the variations of the medium to be extremely small, we may neglect the reaction and calculate the aggregate reflexion as if the primary waves were undisturbed. The partial reflexion which takes place at x is represented by
9—2 which the reader should refer.]bably on account of the less effective rubbing and wiping near the closed end. But what exactly is involved in rubbing and wiping ? I ventured to suggest before that possibly grease may penetrate the glass somewhat. From such a situation it might not easily be removed, or, on the other hand, introduced.